Modelling of an Ionic Electroactive Polymer by the Thermodynamics of Linear Irreversible Processes
aa r X i v : . [ phy s i c s . c o m p - ph ] J a n Modelling of an Ionic Electroactive Polymer bythe Thermodynamics of Linear IrreversibleProcesses
M. Tixier and J. Pouget
Abstract
Ionic polymer-metal composites consist in a thin film of electro-activepolymers (Nafion for example) sandwiched between two metallic electrodes. Theycan be used as sensors or actuators. The polymer is saturated with water, whichcauses a complete dissociation and the release of small cations. The strip undergoeslarge bending motions when it is submitted to an orthogonal electric field and viceversa. We used a continuous medium approach and a coarse grain model; the sys-tem is depicted as a deformable porous medium in which flows an ionic solution. Wewrite microscale balance laws and thermodynamic relations for each phase, then forthe complete material using an average technique. Entropy production, then con-stitutive equations are deduced : a Kelvin-Voigt stress-strain relation, generalizedFourier’s and Darcy’s laws and a Nernst-Planck equation. We applied this model toa cantilever E.A.P. strip undergoing a continuous potential difference (static case);a shear force may be applied to the free end to prevent its displacement. Appliedforces and deflection are calculated using a beam model in large displacements. Theresults obtained are in good agreement with the experimental data published in theliterature.
Electro-active polymers (EAP) have attracted much attention from scientists and en-gineers because of their very promising applications in many areas of science and
Mireille TixierD´epartement de Physique, Universit´e de Versailles Saint Quentin, 45, avenue des Etats-Unis, F-78035 Versailles, France, e-mail: [email protected]
Jo¨el PougetSorbonne Universit´e, UPMC Univ. Paris 6, UMR 7190, Institut Jean le Rond d’Alembert, F-75005Paris, France; CNRS, UMR 7190, Institut Jean le Rond d’Alembert, F-75005 Paris, France, e-mail: [email protected] the growing market. Their behavior and electro-chemical-mechanical interactionsare of great interest and curiosity for research. In particular, the properties of thesematerials are highly attractive for biomimetic applications (for instance, in roboticmechanisms are based on biologically inspired models), for the rise of artificial mus-cles (Bar-Cohen, 2001), and for haptic actuators. More recently EAPs are excellentcandidates for energy harvesting devices (Brafau-Penella et al., 2008). Promisingapplications of this smart material consisting of micro-electromechanical systems(MEMS) at the sub-micron scale are also investigated for accurate medical control(Yoon et al., 2007).The purpose of the present study is to construct step by step a micro-mechanicalmodel which accounts for couplings between the ion transport, electric field andelastic deformation in order to deduce the constitutive equations for this material.Next, an application to the actuation of beam made of thin layer of EAP is presented.Roughly speaking, an electro-active polymer (EAP) is a polymer that exhibits amechanical response, such as stretching, contracting, or bending for example, whensubject to an electric field (only few volts are needed for actuation). Conversely, theEAP can produce energy in response to a mechanical loading.The terminology electro-active polymer has very wide meaning and can be appliedto a large category of materials. The electro-active polymers are generally divided intwo main classes : (i) the electronic EAPs, in which activation is caused by electro-active force between both electrodes which squeezes the polymer and (ii) the ionicEAPs, in which actuation is due to the displacement of ions inside the polymer. Bothclasses are divided into subfamilies according to the physical or chemical principlesof activation. The electronic EAP family encompasses ferroelectric polymers, elec-trets (PolyVinyliDene Fluoride (PVDF) is an example), dielectric elastomers, elec-troactive papers, liquide crystal polymers and many others. The ionic EAP categorycomprises ionic gels, ionic composite (IPMC) (such as Nafion or Flemion), ionicconductor polymers (the strong conductivity is due to oxychloreduction process),nanotube of carbone (the electrolyte is modified by additional charges which pro-duce volume change), electrorheologic fluid (fluid with micro particules changingthe rheological properties of fluid, viscosity for instance) among others. The readercan refer to Bar-Cohen (2001) for many more details. Each category possesses theirown advantages and their drawbacks.The present work addresses investigation of electro-active polymers belonging toionic class and more precisely to ionic-exchange polymer-metal composite (IPMCbecause of the metallic electrodes on the layer faces). The latter consists in an ionicpolymer (Nafion, for instance) sandwiched with two electrodes onto the upper andlower surfaces of the polymer layer. Katchalsky, on 1949 was one of the first inves-tigators to report the ionic chemo-mechanical deformation of polyectrolytes suchas polyacrylic acid or polyvinyl chloride systems. More recently a great interesthas been devoted to EAPs due to the similarities with biological tissues in terms ofachievable stress and they are often called artificial muscles. Moreover, the materialhas potential applications in the field of robotics, medical technology and so on. odelling of an ionic electroactive polymer 3
Investigation on EAP have been traced to Shahinpoor and co-workers (Shahinpoor,1994; Shahinpoor et al., 1998) and some many other researchers.Modeling of EAPs must include complicated electro-chemical-mechanical cou-plings. Different kinds of approaches have been proposed. Newbury and Leo (2001,2002) developed empirical and heuristic models to explain sensing and actuatingproperties of ionic polymer benders. Model based on electrostatic interactions pro-duced by ion motion has been developed by Todokoru et al. (2000). A model in-cluding the effect of electric field, current, pressure gradient and water flux as statevariables has been proposed by de Gennes et al. (2000) using the concept of irre-versible thermodynamics. A more sophisticated model nonetheless closer to the re-alistic properties was developed by Nemat-Nasser and Li (2000) and Nemat-Nasser(2002). The model is based on the micro-mechanics of ionic polymers includingion transport. Finite element 3D model has been studied by Vokoun et al. (2015) tosolve the basic governing physical equations for EAPs proposed by Nemat-Nasserwith given boundary conditions. A model of electro-viscoelastic polymers as an ex-tension of the nonlinear electro-elasticity theory has been discussed by Ask et al.(2012) and finite element numerical simulations were also presented.The proposed model accounts for electro-mechanical and chemical-electric cou-pling of ion transport, electric field and elastic deformation to produce the responseof the EAP. We first investigate the conservation laws of the different phases at themicro level. An averaging statistical process applied to the different phase quan-tities and to the equations of the conservation laws at the micro-scale is used todeduce, in a representative elementary volume containing all the phases, the equa-tions of the conservation laws of the polymer at the continuum level. We write downconservation laws (mass, momentum, energy) in the framework of non-equilibriumthermodynamics. The thermodynamics of linear irreversible process allows us toidentify the fluxes and generalized forces (Tixier and Pouget, 2014) and the con-stitutive equations for the continuum model are consequently obtained (Tixier andPouget, 2016). A generalized Darcy’s law and the balance for ion flux (a kind ofNernst-Plank equation) are deduced from the thermodynamics relations and Gibbsrelation. Along with the former equations the stress-strain equation and that of theelectric charge conservation are also considered (Tixier and Pouget, 2016).The paper is organized as follows, the next section is devoted to the description ofthe EAP giving the main properties and the way of modeling the material. The sec-tion reports also the method used for arriving at the continuum model. The Sect. 3concerns the conservation laws, especially the energy balance laws. The fundamen-tal thermodynamic relations as well as the entropy production are reported in Sect.4.Gibbs’, Euler’s and Gibbs-Duhem’s relations are given in order to deduce the rate ofentropy production. The constitutive equations are presented in Sect. 5, in particu-lar the stress-strain relation, Nernst-Plank equation and generalized Darcy’s law arewritten in terms of concentration, electric field and pressure. The Sect. 6 proposes avalidation of the model by studying the bending actuation of EAP layer subject toa constant difference of electric potential applied to the upper and lower electrodes.
M. Tixier and J. Pouget
Comparisons to experimental results available in the literature ascertain the presentmodel. The most pertinent results are summarized in the conclusions.
The system we study is an ionic polymer-metal composite (IPMC); it consists of amembrane of polyelectrolyte coated on both sides with thin metal layers acting aselectrodes. The polymer is saturated with water, which results in a quasi-completedissociation : anions remain bound to the polymer backbone, whereas small cationsare released in water (Chab´e, 2008). When an electric field perpendicular to theelectrodes is applied, cations are attracted by the negative electrode and carry solventaway by osmosis. As a result, the polymer swells near the negative electrode andcontracts on the opposite side, leading to the bending of the strip.
Fig. 1
Deformable porous medium : (a) Undeformed strip (b) Strip bending under an appliedelectric field
To model this system, the polymer chains are assimilated to a deformable porousmedium saturated by an ionic solution composed by water and cations. We sup-pose that the solution is dilute. We depicted the complete material as the superposi-tion of three systems : a deformable solid made up of polymer backbone negativelycharged, a solvent (the water) and cations (for schematic representation, see the in-sets a and b of Fig. 1). The solid and liquid phases are assumed to be incompressiblephases separated by an interface whose thickness is supposed to be negligible. Weidentify the quantities relative to the different components by subscripts : 1 refersto cations, 2 to solvent, 3 to solid, i to the interface and 4 to the solution, that isboth components 1 and 2; quantities without subscript refer to the whole material.The different components (except 1) as well as the global material are assimilatedto continua. We venture the hypothesis that gravity and magnetic field are negligi-ble, so the only external force acting on the system is the electric force (Tixier andPouget, 2014). odelling of an ionic electroactive polymer 5 We describe this medium using a coarse-grained model developed for two-phasemixtures (Drew, 1983; Drew and Passman, 1998; Ishii and Hibiki, 2006; Lhuillier,2003; Nigmatulin, 1979, 1990). We use two scales. The microscopic scale must besmall enough so that the corresponding volume only contains a single phase (3 or4), but large enough to use a continuous medium model. For Nafion completelysaturated with water, it is about hundred Angstroms. At the macroscopic scale, wedefine a representative elementary volume (R.E.V.) which contains the two phases;it must be small enough so that average quantities relative to the whole materialcan be considered as local, and large enough so that these averages are relevant.Its characteristic length is about micron (Chab´e, 2008; Collette, 2008; Gierke et al.,1981).A microscale Heaviside-like function of presence χ k ( −→ r , t ) has been defined forthe phases 3 and 4 χ k = when phase k occupies point −→ r at time t , χ k = otherwise . (1)The function of presence of the interface is the Dirac-like function χ i = −−→ ∇ χ k · −→ n k (in m − ) where −→ n k is the outward-pointing unit normal to the interface in the phase k . hi k denotes the average over the phase k of a quantity relative to the phase k only.The macroscale quantities relative to the whole material are obtained by statisticallyaveraging the microscale quantities over the R.E.V., that is by repeating many timesthe same experiment. We suppose that this average, denoted by hi , is equivalentto a volume average (ergodic hypothesis) and commutes with the space and timederivatives (Leibniz’ and Gauss’ rules; Drew, 1983; Lhuillier, 2003). A macroscalequantity g k verifies g k = (cid:10) χ k g k (cid:11) = φ k (cid:10) g k (cid:11) k , (2)where g k is the corresponding microscale quantity and φ k = h χ k i the volume frac-tion of the phase k . g k is relative to the total volume of the whole material. In thefollowing, we use superscript to indicate microscale quantities; the macroscalequantities, which are averages defined all over the material, are written without su-perscript. In practice, contact area between phases 3 and 4 has a certain thickness; extensivephysical quantities vary from one bulk phase to the other one. This complicatedreality can be modelled by two uniform bulk phases separated by a discontinuitysurface Σ whose localization is arbitrary. Let Ω be a cylinder crossing Σ , whosebases are parallel to Σ . We denote by Ω and Ω the parts of Ω respectively includedin phases 3 and 4. M. Tixier and J. Pouget
The continuous quantities relative to the contact zone are identified by a super-script and no subscript. A microscale quantity per surface unit g i related to theinterface is defined by g i = lim Σ −→ Σ (cid:26) Z Ω g dv − Z Ω g dv − Z Ω g dv (cid:27) , (3)where Ω and Ω are small enough so that g and g are constant. Its average overthe R.E.V. is the volume quantity g i defined by g i = (cid:10) g i χ i (cid:11) . (4)We arbitrarily fix the interface position in such a way that it has no mass density.The different phases do not interpenetrate, thus we can write on the interfaces −→ V = −→ V = −→ V = −→ V = −→ V i , (5)where −→ V k denotes the local velocity of the phase k . Moreover, we neglect all thevelocities fluctuations on the R.E.V. scale. In order to write the balance equations, it is necessary to calculate the variationsof the extensive quantities following the material motion. This raises a problembecause the different phases do not move with the same velocity : velocities of thesolid and the solution are a priori different. Let us consider an extensive quantityof density g ( −→ r , t ) relative to the whole material. We can define particle derivativesfollowing the motion of the solid ( d dt ) , the solution ( d dt ) or the interface ( d i dt ) d k gdt = ∂ g ∂ t + ∇ g · −→ V k . (6)According to the theory developped by O. Coussy (Biot, 1977; Coussy, 1989,1995), we are also able to define a derivative following the motion of the differentphases of the medium. We will call it the ”material derivative” ρ DDt (cid:18) g ρ (cid:19) = ∑ k = , , i ρ k d k dt (cid:18) g k ρ k (cid:19) = ∑ k = , , i ∂ g k ∂ t + div (cid:16) g k −→ V k (cid:17) , (7)where ρ k is the mass density of the phase k and g , g and g i the densities rela-tive to the total actual volume attached to the solid, the solution and the interface,respectively g = g + g + g i . (8) odelling of an ionic electroactive polymer 7 This derivative must not be confused with the derivative ddt following the barycentricvelocity −→ V ρ DDt (cid:18) g ρ (cid:19) = ρ ddt (cid:18) g ρ (cid:19) − ∑ k h div (cid:16) g k (cid:16) −→ V − −→ V k (cid:17)(cid:17)i . (9) The balance equation of an extensive microscale quantity g k ( −→ x , t ) writes ∂ g k ∂ t + div (cid:18) g k −→ V k (cid:19) = − div −→ A k + B k , (10)where −→ A k is the flux of g k linked to phenomena other than convection and B k thevolume production of g k (source term). At the macroscopic scale ∂ g k ∂ t + div (cid:16) g k −→ V k (cid:17) = − div −→ A k + B k − (cid:28) −→ A k . −→ n k χ i (cid:29) , (11)in which −→ A k = (cid:28) χ k −→ A k (cid:29) , B k = (cid:10) χ k B k (cid:11) . (12)In the case of an interface, the balance law is according to Ishii and Hibiki (2006) ∂ g i ∂ t + div s (cid:18) g i −→ V i (cid:19) = ∑ , (cid:20) g k (cid:18) −→ V k − −→ V i (cid:19) . −→ n k + −→ A k . −→ n k (cid:21) − div s −→ A i + B i , (13)where −→ A i is the surface flux of g i along the interface due to phenomena other thanconvection, B i the surface production of g i and div s the surface divergence operator.At the macroscopic scale ∂ g i ∂ t + div (cid:16) g i −→ V i (cid:17) = ∑ , (cid:28) χ i −→ A k . −→ n k (cid:29) − div −→ A i + B i . (14)Note that g i , −→ A i and B i are volume quantities.For the complete material, we deduce by summation : ρ DDt (cid:18) g ρ (cid:19) = − div −→ A + B , (15)where −→ A = ∑ , , i −→ A k , B = ∑ , , i B k . (16) M. Tixier and J. Pouget
We now write the balance laws of the following quantities : mass, electric charge,linear momentum, the different energies (potential, kinetic, total and internal), en-tropy as well as Maxwell’s equations.
In the absence of chemical reaction, there is no volume production of mass and theonly flux is due to convection. The macroscale mass continuity equation thus writes ∂ρ k ∂ t + div (cid:16) ρ k −→ V k (cid:17) = ( k = , ) , ∂ρ∂ t + div (cid:16) ρ −→ V (cid:17) = , (17)with ρ = φ CM , ρ k = φ k ρ k ( k = , ) , ρ = ρ + φ CM , (18) M k is the molar mass of the component k and C the cations molar concentrationrelative to the liquid phase. We assume the concentration fluctuations are negligible. As in the case of mass, there is, for the electric charge, neither source term nor fluxexcept convection. The electric charge conservation law can be written div −→ I + ∂ρ Z ∂ t = , (19)where ρ Z = ∑ , ρ k Z k + Z i , −→ I = ∑ , , i −→ I k , (20)and −→ I = ρ Z −→ V , −→ I = ρ Z −→ V . (21) Z k denotes the electric charge per unit of mass and −→ I k the current density vector ofphase or interface k .The electric field −→ E k and the electric displacement −→ D k verify Maxwell’s equationsand their boundary conditions. We assume that the electric field fluctuations arenegligible and that macroscale electric fields are identical in all the phases. We thusobtain Maxwell’s equations for the complete material −→ rot −→ E = −→ , div −→ D = ρ Z , −→ D = ε −→ E , (22) odelling of an ionic electroactive polymer 9 where the permittivity ε of the whole material writes ε = ∑ , φ k (cid:10) ε k (cid:11) k . (23)We conclude that the E.A.P. behaves like an isotropic homogeneous linear dielectric.However its permittivity varies over time and space because of variations of thevolume fractions φ k . Since the gravity and the magnetic field are negligible, the only force applied isthe electric force (source term). The linear momentum flux is related to the stresstensors σ e k of the different phases. From (5), we deduce that the linear momentumof the interface is zero, which leads to the following linear momentum conservationlaw for the interfaces ∑ , D σ k f · −→ n k χ i E = Z i −→ E i . (24)We obtain for the complete material ρ D −→ VDt = −→ div σ e + ρ Z −→ E = −→ div (cid:20) σ e + ε (cid:18) −→ E ⊗ −→ E − E e(cid:19)(cid:21) + E −−→ grad ε , (25)using Maxwell’s equations (22). 1 e denotes the identity tensor. σ e = ∑ k = , σ k e , (26)is a symmetric tensor. ε (cid:16) −→ E ⊗ −→ E − E e(cid:17) is the Maxwell’s tensor, which is heresymmetric. The additional term E −−→ grad ε results from the non homogeneous mate-rial permittivity. Potential energy production is equal to the volume power −−→ E k · −→ I k of the force dueto the action of the electric field on the electric charges. The two phases are supposedto be non-dissipative isotropic linear media. As a consequence the balance equationfor the potential energy (Poynting’s theorem) can be written in the integral form(Jackson, 1975; Maugin, 1988) ddt Z Ω (cid:16) −→ E · −→ D + −→ B · −→ H (cid:17) dv = − I ∂Ω (cid:16) −→ E ∧ −→ H (cid:17) · −→ n ds − Z Ω −→ E · −→ I dv . (27)The left hand side represents the variation of the potential energy attached to thevolume Ω following the charge motion. If the charges are mobile, the associatedlocal equation is written as follows for the phase k , neglecting the magnetic field ∂ E pk ∂ t + div (cid:18) E pk −→ V k (cid:19) = −−→ E k · −→ I k k = , , (28)in which E pk = −→ D k · −→ E k is the potential energy per unit of volume of the phase k .The potential energy balance equation for the whole material is then ρ DDt (cid:18) E p ρ (cid:19) = −−→ E · −→ I , (29)where E p = −→ D · −→ E . The relative velocities of the different phases are negligible compared to the ve-locities measured in the laboratory reference frame. Let’s take for example a stripof Nafion which is µ m thick and long, bending in an electric field;the tip displacement is about and is obtained in (Nemat-Nasser and Li,2000). Relative velocities of the order of − m s − and absolute velocities closeto − m s − are thus obtained (Tixier and Pouget, 2016). In a first approxima-tion, we can identify the kinetic energy E c of the complete material and the sum ofthe kinetic energies of the constituents. The kinetic energy balance equation derivesfrom the linear momentum balance equation ρ DDt (cid:18) E c ρ (cid:19) = ∑ , h div (cid:16) σ k e · −→ V k (cid:17) − σ k e : grad g −→ V k i + (cid:16) −→ I − −→ i (cid:17) · −→ E , (30)where the diffusion current of the cations in the solution and of the interfaces is −→ i = −→ I − ∑ k = , (cid:16) ρ k Z k −→ V k (cid:17) − Z i −→ V i . (31) The total energy E is the sum of internal, potential and kinetic energies. The sourceterm is null, and the flux comes from contact forces work and heat conduction −→ Q .The total energy conservation law for the whole material is odelling of an ionic electroactive polymer 11 ρ DDt (cid:18) E ρ (cid:19) = div ∑ k = , σ k e · −→ V k ! − div −→ Q . (32) Internal energy U is the difference between total energy and potential and kineticenergies U = E − E c − E p , (33)which leads to ρ DDt (cid:18) U ρ (cid:19) = ∑ , (cid:16) σ k e : grad g −→ V k (cid:17) + −→ i · −→ E − div −→ Q . (34) The energy balance laws we have written are relative to a thermodynamic closedsystem because of the use of the material derivative. Source terms correspond toconversion of one kind of energy into another one. Energy exchanges are summa-rized in the table 1. Fluxes can be considered as the rate of variation of the quantityassociated with the conduction phenomenon. The kinetic energy flux is equal to thework of the contact forces, the internal energy flux is the heat flux and the totalenergy flux is the sum of the two. We verify that there is no source term in thislast equation. (cid:16) −→ I − −→ i (cid:17) · −→ E results in a motion of the electric charges subject tothe electric field and is a conversion of potential energy into kinetic energy. −→ i · −→ E can be seen as Joule heating, which is a conversion of potential energy into internalenergy. ∑ , (cid:16) σ k e : grad g −→ V k (cid:17) represents the viscous dissipation, which converts kineticenergy into heat. Table 1
Energy exchangesflux E c ←→ E p U ←→ E p E c ←→ UE p − (cid:16) −→ I − −→ i (cid:17) · −→ E −−→ i · −→ EE c ∑ , σ k f · −→ V k + (cid:16) −→ I − −→ i (cid:17) · −→ E − ∑ , σ k f : grad g −→ V k U −−→ Q + −→ i · −→ E + ∑ , (cid:16) σ k f : grad g −→ V k (cid:17) E ∑ , σ k f · −→ V k − −→ Q We shall now write the thermodynamic relations of the electroactive polymer. Thethermodynamics of linear irreversible processes will allow us to identify the fluxesand the generalized forces (Tixier and Pouget, 2016).
The entropy balance law of the whole material is written as ρ DDt (cid:18) S ρ (cid:19) = s − div −→ Σ , (35)where S , −→ Σ and s denote the entropy, the entropy flux vector and the rate of entropyproduction, respectively. According to de Groot and Mazur (1962), the Gibbs relations of a single constituentsolid and of a two-constituent fluid can be written at the microscopic scale ρ d dt (cid:18) U ρ (cid:19) = p
03 1 ρ d ρ dt + σ e g s : d ε e s dt + ρ T d dt (cid:18) S ρ (cid:19) , d dt (cid:16) U ρ (cid:17) = T d dt (cid:16) S ρ (cid:17) − p d dt (cid:16) ρ (cid:17) + ∑ k = , µ k d dt (cid:16) ρ ′ k ρ (cid:17) , (36)where T k denotes the absolute temperature, ε e and σ e g the strain tensor and theequilibrium stress tensor, ε s f and σ es g the strain and stress deviator tensors. Thesolid pressure p is related to the stress tensor and verifies the Euler’s relation, aswell as the liquid pressure p p = − tr (cid:16) σ e g(cid:17) = T S − U + µ ρ , U − T S + p = µ CM + µ ρ φ φ , (37) µ k is the chemical potential per unit of mass. We venture the hypothesis that thefluctuations over the R.E.V. of the intensive thermodynamic quantities (pressures,temperatures and chemical potentials) and of the strain and equilibrium stress ten-sors are negligible. We also suppose that the solid deformations are small. Makingthe hypothesis of local thermodynamic equilibrium, we derive odelling of an ionic electroactive polymer 13 p = p = p = p = p , T = T = T = T i = T = T . (38)We thus obtain Gibbs, Euler’s and Gibbs-Duhem relations of the whole material T DDt (cid:16) S ρ (cid:17) = DDt (cid:16) U ρ (cid:17) + p DDt (cid:16) ρ (cid:17) − ρ σ e f s : d ε e s dt − ∑ , µ k ρ ρ d dt (cid:16) ρ k ρ (cid:17) , p = T S − U + ∑ k = , , µ k ρ k , dpdt = S dTdt + ∑ k = , , ρ k d µ k dt − σ e f s : grad g −→ V . (39) The stress tensor is composed of the equilibrium stress tensor σ e f and the viscousstress tensor σ v f ; this second part vanishes at equilibrium σ e = σ e f + σ v f = − p e + σ e f s + σ v f . (40)Combining the internal energy and entropy equations with the Gibbs relation, therate of entropy production s can be identified s = T σ v f : grad g −→ V + T −→ i ′ ·−→ E − T −→ Q ′ ·−−→ gradT + ∑ k = , , ρ k (cid:16) −→ V − −→ V k (cid:17) ·−−→ grad µ k T , (41)with −→ i ′ = −→ I − ρ Z −→ V , −→ Q ′ = −→ Q − ∑ k = , U k (cid:16) −→ V − −→ V k (cid:17) − ∑ k = , σ k e · (cid:16) −→ V k − −→ V (cid:17) . (42) We define the mass diffusion flux of the cations in the solution −→ J and the massdiffusion flux of the solution in the solid −→ J −→ J = ρ (cid:16) −→ V − −→ V (cid:17) , −→ J = ρ (cid:16) −→ V − −→ V (cid:17) . (43)These two fluxes are linearly independant. The diffusion current −→ i ′ and the fluxes ρ k (cid:16) −→ V k − −→ V (cid:17) can be expressed as functions of these two fluxes. We thus identify thefluxes along with the associated generalized forces (table 2). Table 2
Generalized fluxes and forcesFluxes Forces tr σ v f T div −→ V −→ Q ′ −−→ grad T −→ J ρ ρ h T Z −→ E − −−→ grad µ T + −−→ grad µ T i −→ J ρ ρ h T (cid:16) ρ ρ Z − Z (cid:17) −→ E − ρ ρ −−→ grad µ T − ρ ρ −−→ grad µ T + −−→ grad µ T i σ v f s T h (cid:16) grad g −→ V + grad g −→ V T (cid:17) − (cid:16) div −→ V (cid:17) ei We venture the hypothesis that the medium is isotropic. According to Curie sym-metry principle, there can not be any coupling between fluxes and forces whosetensorial ranks differs from one unit. Moreover, we suppose that coupling betweenfluxes and different tensorial rank forces are negligible, which is commonly admit-ted (de Groot and Mazur, 1962). We thus obtain a tensorial law (the rheologicalequation) and three vectorial constitutive equations.
Considering the symmetry of the tensor σ v f , the scalar and tensorial fluxes are linearfunctions of the corresponding forces. Assuming that the complete material satisfiesthe Hooke’s law at equilibrium and the liquid phase is newtonian and stokesian, thepressure verify p = − tr (cid:16) σ e f(cid:17) = (cid:18) λ + G (cid:19) tr ε e , (44)where λ and G denote the first Lam´e constant and the shear modulus of the completematerial, respectively, and ε e the material strain ε e = (cid:16) grad g −→ u + grad g −→ u T (cid:17) and • ε e = (cid:16) grad g −→ V + grad g −→ V T (cid:17) . (45) −→ u is the displacement vector. The stress tensor of the complete material thus iden-tifies with a Kelvin-Voigt model σ e = λ (cid:0) tr ε e(cid:1) e + G ε e + λ v tr • ε e e + µ v • ε e , (46) λ v and µ v are viscoelastic coefficients.The elastic coefficients have the following values (Bauer et al., 2005; BarclaySatterfield and Benziger, 2009; Silberstein and Boyce, 2010), which are in agree-ment with the usual ones odelling of an ionic electroactive polymer 15 G ∼ . Pa , λ ∼ Pa , E ∼ . Pa , ν ∼ . , (47)where E is the Young’s modulus and ν the Poisson’s ratio. Viscoelastic coefficientscan be deduced from uniaxial tension tests (Barclay Satterfield and Benziger, 2009;Silberstein and Boyce, 2010; Silberstein et al., 2011) and relaxation times for atraction, which are close to 15 s (Silberstein, 2008; Silberstein and Boyce, 2010;Silberstein et al., 2011) λ v ∼ Pa s , µ v ∼ Pa s . (48)These coefficients depend strongly on the solvent concentration and on the temper-ature, especially if it is close to the glass transition. Vectorial constitutive equations require nine phenomenogical coefficients, which area priori second-rank tensors; they can be replaced by scalars because of the isotropyof the medium. The first equation that we obtain is a generalized Fourier’s law. Wewill approximate the two other by restricting ourselves to the isothermal case andfocusing on a particular E.A.P. : Nafion 117 Li + .The liquid phase is a dilute solution of strong electrolyte. According to Diu et al.(2007), mass chemical potentials can be written on a first approximation µ ( T , p , x ) ≃ µ ( T , p ) + RTM ln (cid:16) C M ρ (cid:17) , µ ( T , p , x ) ≃ µ ( T , p ) − RT ρ C , µ ( T , p , x ) = µ ( T ) , (49)where R = , J K − is the gas constant. µ and µ denote the chemical potentialsof the single solid and solvent, and µ depends on the solvent and the solute. TheGibbs-Duhem’s relations for the solid and the liquid phases enable the calculationof −−→ grad µ k −−→ grad µ = − S ρ −−→ gradT + v M −−→ grad p + RT ρ M M C −−→ grad (cid:16) CM ρ (cid:17) , −−→ grad µ = − S ρ −−→ gradT + v M −−→ grad p − RTM −−→ grad (cid:16) CM ρ (cid:17) , −−→ grad µ = − S ρ −−→ gradT , (50)where v k denotes the partial molar volume of the constituent k .The physicochemical properties of the Nafion are well documented. Its equiv-alent weight, that is to say, the weight of polymer per mole of ionic sites is M eq ∼ . kg eq − (Gebel, 2000). The solution volume fraction φ is close to 38%(Cappadonia et al., 1994; Choi et al., 2005). Other parameters are resumed in ta-ble 3 (Heitner-Wirguin, 1996; Nemat-Nasser and Li, 2000). The anions molar con- centration, which is equal to the average cations concentration, is C moy = ρ φ M eq φ ∼ . mol m − . The dynamic viscosity of water is η = − Pa s . We deduce themass densities of the complete material ρ ∼ . kg m − . We moreover supposethat the temperature is T = K . The electric field is typically about 10 V m − (Nemat-Nasser and Li, 2000). Table 3
Nafion 117 Li + parametersCations Solvent Solid M k (cid:0) kg mol − (cid:1) −
18 10 − - ρ k (cid:0) kg m − (cid:1) .
08 10 v k (cid:0) m mol − (cid:1) M ρ ∼ −
18 10 − ρ k (cid:0) kg m − (cid:1) . . Z k (cid:0) C kg − (cid:1) . Considering this numerical estimations, we can write in a first approximation Z >> Z , ρ ∼ ρ ∼ ρ >> ρ , ρ Z ∼ ρ Z . (51) It is commonly accepted that the non-diagonal phenomenological coefficients aresmall compared to the diagonal ones. The expression of the mass diffusion flux ofthe cations J can be identified with a Nernst-Planck equation (Lakshminarayanaiah,1969) −→ V = − DC (cid:20) −−→ gradC − Z M CRT −→ E + Cv RT (cid:18) − M M v v (cid:19) −−→ grad p (cid:21) + −→ V . (52) D ∼ − m s − denotes the mass diffusion coefficient of the cations in the liquidphase (Zawodsinski et al., 1991). This equation expresses the equilibrium of an ionsmole under the action of four forces : the Stokes friction force, which is propor-tional to −→ V − −→ V , the pressure force, the electric force and the thermodynamic force − M −−→ grad µ .The order of magnitude of the different terms of this equation can be estimated.According to Farinholt and Leo (2004) and Nemat-Nasser (2002), the concentrationgradient verifies (cid:12)(cid:12)(cid:12) −−→ gradC (cid:12)(cid:12)(cid:12) - mol m − . (53)The pressure gradient can be roughly estimated using the Darcy’s law; it is about10 Pa m − . We thus obtain odelling of an ionic electroactive polymer 17 M CRT Z (cid:12)(cid:12)(cid:12) −→ E (cid:12)(cid:12)(cid:12) ∼ . mol m − , Cv RT (cid:16) − M M v v (cid:17) (cid:12)(cid:12)(cid:12) −−→ grad p (cid:12)(cid:12)(cid:12) ∼ . mol m − . (54)The electric field and the mass diffusion have the leading effects; we thereafter ne-glect the pressure gradient term. The expression of the mass diffusion flux of the solution in the solid J identifieswith a generalized Darcy’s law −→ V − −→ V ≃ − K η φ h −−→ grad p − ρ ( Z − Z ) −→ E i − RM C ρ L −−→ gradC . (55)where L is a phenomenological coefficient and K the intrinsic permeability of thesolid phase, which is on the order of the square of the pore sizes, that is 10 − m .The orders of magnitude of the different terms are K η φ (cid:12)(cid:12)(cid:12) −−→ grad p (cid:12)(cid:12)(cid:12) ∼ . − m s − , K η φ ρ ( Z − Z ) (cid:12)(cid:12)(cid:12) −→ E (cid:12)(cid:12)(cid:12) ∼ . m s − , RM C ρ L (cid:12)(cid:12)(cid:12) −−→ gradC (cid:12)(cid:12)(cid:12) << . − m s − . (56)The latter term can therefore be neglected. The first term represents the mass pres-sure force and the second one is the mass electric force; it expresses the motionof the solution under the action of the electric field and reflects an electro-osmoticphenomenon.The distribution of cations becomes very heterogeneous (Farinholt and Leo,2004; Nemat-Nasser, 2002) : they gather near the negative electrode, where Z >> Z ; the expression obtained coincides with that of Biot (1955). Near the positiveelectrode, where the cation concentration is very low, Z << Z ; this corresponds tothe result obtained by Grimshaw et al. (1990) and Nemat-Nasser and Li (2000). Inthe center of the strip, the mass electric force exerted on the solution is proportionalto the net charge ( Z − Z ) . In order to validate the model that we have just described, we apply it to the staticcase of a cantilevered E.A.P. strip bending under the effect of a permanent electricfield. In addition, the strip might undergo the action of a shear force exerted on thefree end in order to prevent its displacement (blocking force).
A continuous constant voltage ϕ is applied between the two faces of the strip. As aconsequence, the partial derivatives with respect to time and the velocities are zero.We focuse on Nafion Li + strip of length L = cm , of thickness 2 e = µ m andof width 2 l = mm subject to a potential difference ϕ = V . We postulate thatthe volume fraction φ is constant; this hypothesis will be checked a posteriori. As aconsequence, the dielectric permittivity of the whole material is a constant too. Weassume that it is equal to that measured by Deng and Mauritz (1992) for a materialvery close to the Nafion : ε ∼ − F m − . Considering the strip dimensions, it isa two-dimensional problem. A coordinate system Oxyz is chosen such that the axis Oz is parallel to the imposed electric field, the axis Ox is along the length of the stripand Oy along its width. We venture the hypothesis that the local electric field axialcoordinate E x is negligible compare to the normal one E z . On a first approximation, C , E z , p , the local electric potential ϕ and the electric charge density ρ Z only dependon the z coordinate. Neglecting the pressure term in (52), we obtain E z = − d ϕ dz , ε dE z dz = ρ Z , dpdz = (cid:0) CF − ρ Z (cid:1) E z , dCdz = FCRT E z , ρ Z = φ F ( C − C moy ) . (57) F = C mol − is the Faraday’s constant. The boundary conditions write lim z →− e ϕ = ϕ , lim z → e ϕ = , R e − e ρ Z d z =
0. (58)The latter condition expresses the electroneutrality condition and is equivalent to E z ( e ) = E z ( − e ) . We introduce the following dimensionless variables: E = E z e ϕ , C = CC moy , ϕ = ϕϕ , ρ Z = ρ Z φ FC moy , p = pF ϕ C moy , z = ze , (59)which leads to E = − d ϕ dz , dEdz = A A ρ Z , dpdz = (cid:0) C + A (cid:1) E , dCdz = A CE , ρ Z = C − . (60) A , A et A are dimensionless constants A = φ e F C moy ε RT ∼ ,
37 10 , A = F ϕ RT ∼ , , A = − ρ Z C moy FC moy ∼ , lim z →− ϕ = , lim z → ϕ = , E ( ) = E ( − ) . (62) odelling of an ionic electroactive polymer 19 We deduce ddz (cid:18) dCCdz (cid:19) = A (cid:0) C − (cid:1) , (63)and C ≃ A exp ( − A ϕ ) , p = CA − A ϕ + Cte . (64)The hydrated polymer can be assimilated to a conductive material. As a conse-quence, the electric field is zero throughout the strip except near the boundaries. Wededuce the values of the different quantities on the sides and at the center of the strip(table 4). The equation (63) can be solved using Matlab. We deduce ρ Z , ϕ , p and Table 4
Boundary values of the physical quantities − Center C A e − A ≃ A ϕ ln A A E r A A h − A ( + ln A ) i r A A h − A ( + ln A ) i p − A A ( − A ln A ) E by (60) and (64). An evaluation of the pressure term of the equation (52) showsthat it does not exceed 2% of the second term of the equation with the nominalconditions chosen. L x z O A p p M pA F p _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ z p O t n x z q q
Fig. 2
Forces exerted on the beam
Given the high deflection values, weused a large displacement beam modelto determine forces, stress and strain.We consider a straight beam clampedat the end O ; the other end A is eitherfree or subject to a shear force bringingthe deflection to a zero value (block-ing force −→ F p ). The polymer is subject toa distributed electric force −→ p p indepen-dent of the x coordinate and orthogonalto the strip. Moreover, the swelling ofthe strip on the side of the negative electrode generates, through the pressure p = σ xx ,a bending moment −→ M pA at the free end of the beam (see figure 2). Considering theelectroneutrality condition, the distributed force and the bending moment verify p p = Z l − l Z e − e ρ ZE z dz dy = l Z e − e ρ ZE z dz = l ε (cid:20) E z (cid:21) e − e = , (65) M pA = Z l − l Z e − e σ xx z dz dy = A Z − p z dz , (66)with A = le F ϕ C moy ∼ , N m . A _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ z R p O t n e x e z q q Fig. 3
Beam on large displace-ments : geometric coordinates
We assume that the Bernoulli and Barr´e SaintVenant hypotheses are verified. Let s and s be thecurvilinear abscissas along the beam respectively atthe rest and deformed configurations, −→ t and −→ n thevectors tangent and normal to the beam, θ the angleof rotation of a cross-section and R p the radius ofcurvature (geometric parameters are given in figure3). It will be assumed that the elongation Λ = dsds is equal to 1. The bending moment in the currentsection is M p = F p ( x − L ) + M pA . (67)The strain tensor is given by ε e = (cid:20)(cid:16) Grad ] −→ u + e(cid:17) T (cid:16) Grad ] −→ u + e(cid:17) − e(cid:21) . (68)In the reference frame linked to the undeformed beam, it follows ε xx = − nR p (cid:18) − n R p (cid:19) ≃ − nR p , (69)where n designates the coordinate in the −→ n direction. Indeed, the beam being thin, | n | << R p . In the case of pure bending, the strain is given by ε xx = M p EI p n where I p = le is the area moment of inertia, with respect to the Oy axis. The shear forcehas a negligible effect on the deflection. We deduce the radius of curvature and theangle of rotation R p = d θ ds = F p EI p ( L − s ) − M pA EI p , θ = F p EI p s ( L − s ) − M pA EI p s , (70)by choosing the point O as the origin of the curvilinear abscissas. The deflection w is obtained by integrating the relation dzds = sin θ . In the case of a cantilever beam( F p = θ = − M pA EI p L = − A L R − p z dz , w = EI p M pA h cos (cid:16) M pA EI p L (cid:17) − i = L A R − p z dz h cos (cid:16) A L R − p z dz (cid:17) − i , (71)with A = L F ϕ C moy eE ∼ , m . The blocking force is the end loading such thatthe deflection is zero odelling of an ionic electroactive polymer 21 Fig. 4
Variation of cation concentration in the thickness of the strip; the distributions at the vicinityof the lower and upper faces (electrodes) are detailled in insets. w ( s = L ) = − Z L sin (cid:20) F p EI p s ( s − L ) + M pA EI p s (cid:21) ds = . (72)Using the Fresnel functions to compute this integral, we obtain for the blockingforce the same result as in small displacements F p = M pA L = A L Z − p z dz . (73) The values obtained for the deflection and the blocking force are in good agree-ment with the experimental values reported in the literature (Nemat-Nasser, 2002;Newbury, 2002; Newbury and Leo, 2002, 2003).The variation of cation concentration in the thickness of the strip is shown infigure 4. This quantity is constant throughout the central part of the strip but variesvery strongly in the vicinity of the electrodes, especially near the negative electrodeon which the cations accumulate. On the contrary, it is noticed near the positiveelectrode an aerea of the order of 0 , µ m without cations. This result is in goodagreement with that of Nemat-Nasser (2002). B l o c k i ng f o r ce ( m N ) Electric potential (V)
Fig. 5
Influence of the potential difference on theblocking force R − p z dz depends only on the im-posed potential ϕ , the thickness e and the cation chosen. The deflec-tion w is therefore independent of thewidth and approximately proportionalto length square, which corresponds toShahinpoor’s measurements (Shahin-poor, 1999). The blocking force is pro-portional to l and inversely propor- tional to L , which is in good agreementwith Newbury and Leo (2003).The deflection varies linearly with the imposed potential difference ϕ , a result inagreement with the experiments of Mojarrad and Shahinpoor (1997) and Shahinpooret al. (1998). The blocking force follows the same tendency (figure 5). We have modeled the behavior of an ionic electroactive polymer saturated withwater and subject to an orthogonal electric field. The presence of water causes acomplete dissociation of the polymer and the release of small cations. We have de-picted this medium as the superposition of three systems with different velocitiesfields : the cations, the solvent and the solid assimilated to a deformable porousmedium. We have used a continuous medium approach and a coarse grain model.We have established the microscale balance equations of mass, linear momentum,energies, entropy and the Maxwell’s equations as well as thermodynamic relationsfor each phase (solid and liquid). We have derived the macroscale equations relativeto the whole material using an average technique and the material derivative con-cept. Thermodynamics of linear irreversible processes have provided the entropyproduction and the constitutive equations of the complete material. We have ob-tained a rheological law of the Kelvin-Voigt type, generalised Fourier’s and Darcy’slaws and a Nernst-Planck equation.We have applied this model to a cantilevered Nafion strip subject to a continuousvoltage between its two faces, which is a static case. The other end may be eitherfree or subject to a shear force preventing its displacement (blocking force). Wehave used a beam model in large displacement. We have drawn the profile of cationsconcentration and evaluated the deflection and the blocking force. We have observedthat the concentration varies very strongly in the vicinity of the electrodes, which ischaracteristic of a conductive material behavior. Our results are in good agreementwith the experimental data published in the literature.To improve this model, we intend to take into account the variations of the per-mittivity with the cation concentration. Another way of improvement is to replacethe rheological law with a Zener model that is better suited to viscoelastic polymers. k = , , , , i subscripts respectively represent cations, solvent, solid, solution (wa-ter and cations) and interface; quantities without subscript refer to the whole mate-rial. Superscript denotes a local quantity; the lack of superscript indicates averagequantity at the macroscopic scale. Microscale volume quantities are relative to the odelling of an ionic electroactive polymer 23 volume of the phase, average quantities to the volume of the whole material. Super-script s indicates the deviatoric part of a second-rank tensor, and T its transpose. C , C moy : cations molar concentrations (relative to the liquid phase); D : mass diffusion coefficient of the cations in the liquid phase; −→ D , −→ D k : electric displacement field; e : half-thickness of the strip; E , G , λ , ν : Young’s and shear modulus, first Lam´e constant, Poisson’s ratio; −→ E , −→ E k : electric field; E , E p ( E pk ), E c , U ( U k , U k ) : total, potential, kinetic and internal energy densities; F = C mol − : Faraday’s constant ; −→ F p : blocking force; −→ i ( −→ i ′ ) : diffusion current; −→ I ( −→ I k , −→ I k ) : current density vectors; I p : area moment of inertia; −→ J k : mass diffusion flux; K : intrinsic permeability of the solid phase; l : half-width of the strip; L : length of the strip; M k : molar mass of component k ; M eq : equivalent weight (weight of polymer per mole of sulfonate groups); −→ M p ( −→ M pA ) : bending moment; −→ n k : outward-pointing unit normal of phase k ; p ( p k , p k ) : pressure; −→ Q ( −→ Q ′ ) : heat flux; R = , J K − : gaz constant; R p : radius of curvature of the beam; s : rate of entropy production; S ( S k ) : entropy density; T ( T k , T k ) : absolute temperature; −→ u : displacement vector; v k : partial molar volume of component k (relative to the liquid phase); −→ V ( −→ V k , −→ V k ) : velocity; w : deflection of the beam; Z ( Z k ) : total electric charge per unit of mass; ε , ε k : permittivity; ε e ( ε k e , ε k e ) : strain tensor; η : dynamic viscosity of water; θ : angle of rotation of a beam cross section; λ v , µ v : viscoelastic coefficients; µ k ( µ k ) : mass chemical potential; ρ ( ρ k , ρ k ) : mass density; σ e ( σ k e , σ k f ), σ v f , σ e f ( σ ek f , σ ek g ) : stress tensor, dynamic stress tensor, equilibriumstress tensor; ΣΣΣ : entropy flux vector; φ k : volume fraction of phase k ; ϕ ( ϕ ): electric potential; χ k : function of presence of phase k ; References
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